Laboratory Games and Quantum Behaviour: The Normal Form with a Separable State Space

The subjective expected utility (SEU) criterion is formulated for a particular four-person “labo- ratory game” that a Bayesian rational decision maker plays with Nature, Chance, and an Experi- menter who influences what quantum behaviour is observable by choosing an orthonormal basis in a separable complex Hilbert space of latent variables. Nature chooses a state in this basis, along with an observed data series governing Chance’s random choice of consequence. When Gleason’s theorem holds, imposing quantum equivalence implies that the expected likelihood of any data series w.r.t. prior beliefs equals the trace of the product of appropriate subjective density and likelihood operators.